Leaves

 I looked out upon my lawn covered with leaves and wondered how many there were. The brute force method is to go count them. Lets say I don't have the time or inclination to do that. I could take some smaller areas of the lawn, count the leaves there, and compare them to the whole.

This is the same kind of thing that is done in surveys. DVD sales, TV ratings, unemployment numbers, all survey to get their results. We recently had a presidential election. It took a while, a long while, because they said they were getting to a 99.5%. That doesn't give you certainty of a number. It just means you are 99.5% certain of being within a range of numbers. Almost all nationwide statistics are surveys. When we don't do that - the census every 10 years - it takes months and months to get to everyone.

I'm sure you know how this kind of thing works, but I took the time to do it, so I feel obligated to share.

I (roughly) marked one yard by one yard squares. There were three of them and they averaged having 125 leaves in each. Then I found the area of the entire yard. Nicely, it is pretty much a trapezoid. The bases about 7 and 14 yards with a height of 7 yards. That gives an area of 73.5 square yards.

Next, a proportion would be 125/1 = x/73.5. So, x = 9187.5 leaves. 

So, a nice problem for maybe an Algebra I or general math class. It can lead into the idea of the surveys that go on every day.

The Lighthouse at Alexandria

I was watching a video about the Lighthouse at Alexandria. It no longer exists, but was one of the seven wonders of the world. The video said it could be seen from thirty miles away. They didn't mention how tall it was. That can be figured out, though. There is a formula. To the right is a diagram of the situation.

Suppose v is the height of the lighthouse. OH is the distance to the horizon from the top of the lighthouse. Using a geometry thereom, the diameter squared = (diameter+h)h. You could also come up with a formula by using the Pythagorean Theorem. In that case, (v+r)squared = r squared + OH squared. And then the radius or diameter of the earth could be substituted in. Another extension could be to find formulas for distance to the horizon for other other planets. Formulas that are then derived usually are simplified by the fact that the value of v is so small compared to r. 

Those formulas can be found on-line. Using one of them, by my calculation, to have the horizon be 30 miles away you would have to be 605 feet up in the air. Pretty tall. The Eiffel Tower is 984 feet. But the tallest now are less than 300 feet. So the video, or I seem to be off. But maybe not. It's certainly possible. You don't get picked as a wonder of the world if you're not pretty impressive.

Evaporation

When I was a lad, I remember looking a drops of rain that had plopped on the sidewalk. It was a light rain so I could make out the individual drops. They gradually evaporated. I noticed that if I used my finger and spread the raindrops, out they evaporated faster. At my tender age I had no idea why. Still not 100% certain, but my guess now is that if you have a drop of water it is losing molecules, i.e., evaporating, from its surface. If you take a drop of water, it is evaporating at a certain rate. If you separate that drop into two drops, it will evaporate faster because there is much more surface area for which it can use to evaporate.

So, let's examine this as a math application. Suppose a drop of water is spherical and has a volume of 10 whatevers. Then suppose we separate that drop into two drops of volume 5 whatevers each. Then we look at their total surface areas. Will they come out the same?

We start with the fact that it has a volume of 10:  (4/3)πr³ = 10
If we solve for r we get a value of r = 1.3365
We can then find its surface area: 4π(1.3365)² = 22.446

Now what if we now have two spheres of 5 each.
Their radii would be:  (4/3)πr³ = 5, so r = 1.0608
The surface area of one drop is 14.140. There are two drops, though, so the total surface area of them would be 28.28

Comparing the two situations, we can in fact state that, since 28.28/22.466 = 1.26, there is 26% more surface area, and so I will postulate a 26% faster drying time for the two drops over the one drop.

That was to be the end of the story, but I thought what if you separate one spherical drop into two? Is it always going to be a 26% greater surface area.

Unfortunately this is beyond my skill level to show all this. You might recall that I only recently found how to write integer exponents. This process involves having the cube root of a fraction all taken to a power of two and other assorted difficulties. So let me map this out leaving a few gaps for you or students to work through. It really is a great problem, though, with the opportunity to review simplifying - some major simplifying.

• Let us say that  (4/3)πr³ = V
• Solve for r
• Substitute this expression into 4πr²
• Now, find the radius for a sphere that his half the original volume:    (4/3)πr³ = (v/2)
• Substitute this r into 4πr²
• Make a ratio of the two radii and simplify 
• You end up with cube root of 16 divided by 2, which is 1.26
• Ta-da. An increase of 26%

Satisfying

The Importance of Proofs

This blog is devoted to showing applications of high school mathematics. At first I thought the following topic wouldn't fit here, but then I decided it does. That is, the idea of proof. It really is tough for students to see the necessity of proofs. Usually encountered mainly in a geometry, they can be a real chore both for students and teachers.

Proofs are hard. So combine that with the fact that what is being proved seems pretty obvious, you get the double whammy of why students hate proofs - they are both hard and pointless.

Do we really need to prove that the sum of the first n odd positive integers is n squared. It wouldn't take to much time to get students to accept that fact without doing a proof. 

The square root of two is irrational. "I found the square root of two on my calculator. The decimals don't repeat. Yes. I'll buy it."

It seems that statements that are true for the first ten or so examples are probably going to end up being true. In fact, it's a little difficult to find situations where that doesn't hold true. If I make a statement that a coin I have will always come up heads, you might have your doubts. If I flip it ten time and I get heads each time you probably think that statement is, in fact, true, because it probably is a two headed coin.

I had heard quite a while ago that there was some statement having to do with prime numbers what was true way down the line, but then turned out not to be false at some point. I never knew exactly what that was, though.  

Well here it is. 

Suppose we want to find out if P is a prime number. Find 2^P and then divide it by P. If the remainder is 2, then P is a prime number. If it is something other than 2, P isn't prime.

I was a doubter at first. I did the first several and they all worked out, though. I'll take one example. Let's try the number 17.

2¹⁷ = 131,072
131,072 divided by 2 is 7,710 remainder 2
Thus 17 is a prime number.

This really works - all the way to 340. Then it doesn't.

2³⁴¹ is so big I can't tell you what it is. I could get my calculator to go as high as to the 331 power, then it refused to do any more.

Anyway, pretty amazing. It works for 1, 2, 3, 4, 5, ..., 340, correctly determining if the number is prime. Then this method claims 341 is prime. But it isn't, because 11 x 31 is 341.

There you go. Proofs aren't pointless after all. Still hard, but not pointless.

            *****************************************************************

Now, about last week's post. We worked out a score for the yet to be played Super Bowl of New England 30, Atlanta 27.

And how did that turn out? Pretty darn close I must say. New England 34, Atlanta 28.

Trapezoids

I always had a bit of a tough time finding examples of trapezoid applications out there in the real world. There are a few, but certainly not as easy as finding shapes such as circles, rectangles, triangles, squares, ... I thought I would go looking and here are a few I found.

The trapezoidal rule is actually from calculus. A little above the geometry level, the basic idea of it would be quite understandable to a geometry student.

Another interesting one is the Mars Rover which contain the Rover's solar panels. I believe they are in the shape of a trapezoid because the panels are initially folded up against the Rover. The trapezoid shape is best for that unfolding transition.

I don't have specific info on most of the others. They're just trapezoids out in the real world.

Snowflakes

"No two snowflakes are alike." You've undoubtedly heard that a time or two. Seemingly, not an important math application. I got to thinking about it and that statement does bring up some important points. 

First it brings up some lessons in basic logic. If we are trying to prove there are not two snowflakes alike, how would be prove or disprove a statement like this? Disproving it could be easy. Find two that are the same. If we could do this, we could put this issue to rest. 

How about proving it to be true. We've all seen lots of pictures of snowflakes. None alike so far. The pioneer in this seems to be a Wilson Bentley from Vermont. He was born in 1865, a time when there weren't a lot of pictures being taken of anything. He had a collection of over 5,000 photographs of snowflakes. He was single (not a surprise) and had plenty of time to devote to his work. None of his matched. The fact that none of them match would not constitute a proof. This would be a good example of inductive reasoning. Here is an opportunity to discuss inductive verses deductive reasoning, and the benefits and drawbacks of each. 

Can deductive reasoning be used here? Let me state that I'm well out of my area here. My lowest grade in high school was a C and that was in Chemistry. I was fine with that since I probably deserved lower. I really did try. Chemistry and I just do not click. Regardless, here we go.

I did some reading to try to figure this out. A water droplet might freeze onto a dust particle. They freeze in a hexagon shape. Most of what I saw kind of glossed over why that is. One statement explaining the snowflake pattern went like this. And I quote:

Hexagonal ice ([1969], ice Ih ⁱ see Phase Diagram), is in Space group P6₃/mmc, 194; symmetry D_6h, Laue class symmetry 6/mmm; analogous to β-tridymite silica or lonsdaleite, having a a six fold screw axis (rotation around an axis in addition to a translation along the axis).

Curse you Chemistry. Anyway a hexagon is formed. Other water molecules latch onto the vertices of the hexagon, growing the snowflake as it falls through the air. Different shapes come about based on the temperature and humidity of the surrounding air. These flakes all take different paths to the ground, thus slightly changing its weather conditions, thus slightly changing the shapes as they grow. These different paths cause different shapes. At any one moment in time, the forming shape has the same weather conditions, giving the snowflake its symmetry. 

One article states that there are 10,000,000,000,000,000,000 molecules of water in a snow flake and they can be rearranged in many different ways. A couple of articles likened this to factorials. The ways to arranged 6 books is 6! (= 720), 7 books is 7! (5,040), and 8 books is 8! is (40,320). I'm not sure finding the number of snowflake designs is as simple taking the factorial of the number of molecules, but I guess their point is that the number of patterns must be huge.

However, I'm still not convinced. Someone estimated there have been approximately 1,000,000,000,000,000,000,000,000,000,000,000,000 snowflakes. Really? No two alike in that bunch. We haven't looked at them all. Even if we did, even with global warming, I'm sure there will be a bunch more. 

In fact, some of the scientists think there might have been duplicates. If the snowflake doesn't have far to fall, and thus doesn't have the chance do grow very much, that greatly increases the chance that two of them could be similar. 

Actually, one scientist claimed she has found a pair. In 1998, Nancy Knight claims she found two alike. I saw a picture and they look pretty convincing. Nerdy scientists, though. have balked at this. Some of the hydrogen atoms (approximately 1 in 3,000) could be deuterium. (Hydrogen usually has just one proton in the nucleus. Deuterium has a proton and a neutron.) This would likely make snowflakes that looked the same, still not be identical.

Some people just can't admit defeat. 

I hope this was helpful. I did my best to be as accurate and thorough as I could. (I'm sorry Mr. Gustafson. I really did try in Chemistry.) 

Statis Pro Baseball One Last Time

I know my blog has been a little heavy with the baseball applications. Specifically with regards to the best game ever made - Statis Pro Baseball. One more week, then I'll move on. This and other older games are great, though, for math applications because its right there in front of you. All the computer games have the statistics / mathematics hidden away in the computer program running it.

This is application is actually from a different game that I played once with a friend of mine. At the time I thought it was kind of ingenious, although I'm not sure I put a lot of thought into how they did it. Each baseball player had a card with spinner which represented statistically what you could expect from him in an at-bat.

If the first batter up was Ty Cobb, I would take his card, spin the spinner and see what he did. It might land on a colored section of card marked "Out". How did they come up with the colors on the cards anyway? Let's make Ty's situation real simple and divide it into sectors for hits and outs. For his career he 4,189 hits in 11,434 times at bat. That makes a batting average of 0.366. This means of course that he gets a hit 36.6 percent of the time which would be a sector of 36.6% of 360 degrees. This is a sector of 131.76 degrees. His chance of going out would be a different colored sector of 360 - 131.76 = 228.24 degrees.

There were more divisions than just hits and outs. Although it has been a while, I'm sure there were singles, doubles, triples, home runs, outs, and walks at least. To build the circle would mean finding percentages, changing them to degrees of a circle, dividing up the circle, and coloring and labeling the sectors. (The picture is not what the spinner looked like, obviously, but that's the idea.)

Whoever came up with it, I thought it was a pretty good game. It also didn't last, but that's the way it goes, I guess.

So, this isn't a high level math application, obviously, but I think an interesting one, and it reviews, protractor use, percentages, and circles.

3D Basketball

I think its been around before this, but I noticed during this year's NBA Finals the use of some kind of 3D technology. It's very impressive. I didn't know how they did that and I still don't, but I have a few leads.

This is possibly not the kind of math application you can share all the details with a class, as its seems a little complicated. In fact, for the NBA to pull this off takes takes several cameras and expensive software.

I first came across some information on how this is done while reading the June 30th issue of Sport Illustrated, "Computers break down the images into voxels (3-D pixels) and the view from any point on the court or field can be re-created into a three-dimensional, 360-degree video of the action."

That didn't tell me a lot, but it did give me the term "voxels" that I could Google.

From WhatIs.com (http://whatis.techtarget.com/definition/voxel) I found this.

"A voxel is a unit of graphic information that defines a point in three-dimensional space. Since a pixel (picture element) defines a point in two dimensional space with its x and y coordinates, a third z coordinate is needed. In 3-D space, each of the coordinates is defined in terms of its position, color, and density. Think of a cube where any point on an outer side is expressed with an x , y coordinate and the third, z coordinate defines a location into the cube from that side, its density, and its color. With this information and 3-D rendering software, a two-dimensional view from various angles of an image can be obtained and viewed at your computer.

Medical practitioners and researchers are now using images defined by voxels and 3-D software to view X-rays, cathode tube scans, and magnetic resonance imaging (MRI) scans from different angles, effectively to see the inside of the body from outside. Geologists can create 3-D views of earth profiles based on sound echoes. Engineers can view complex machinery and material structures to look for weaknesses."

So getting to see NBA replays in 3D is cool, but probably a little trivial. It was interesting to see the application to medicine and geology.

There turned out to be quite a few hits on "voxels". A lot of them dealing with buying the above mentioned software.

Regarding medical applications, the Scientific American website had an interesting article (http://blogs.scientificamerican.com/observations/whats-a-voxel-and-what-can-it-tell-us-a-primer-on-fmri/) on how voxels can be an improvement over regular MRIs.

From another site I found that while "pixel" is short for picture element, "voxel" is short for volume element. That makes sense. Much of the other information I found does not make a lot of sense to me.

From Webopedia.com I found this info: Voxelization is the process of adding depth to an image using a set of cross-sectional images known as a volumetric dataset. These cross-sectional images (or slices) are made up of pixels. The space between any two pixels in one slice is referred to as interpixel distance, which represents a real-world distance.

This article quickly spiraled away from my level of understanding, but this part kind of made sense. For the NBA 3D replay, perhaps each "slice" is the 2D view from a particular camera. Combining this with several other cameras gives various slices. The distances between those slices constitute the third dimension for various points. I don't know, but I might be onto something. 

Next week will be a new topic, or possibly, if I gain any big insights into voxels, maybe Part 2 of this topic.

More Water Towers

Last week we took a look at how water towers work. Certainly students would want to know something about them before they would feel very motivated to do any math with them. Since it took me literally decades before I even had the slightest idea how they worked, I'm assuming most students don't know much about. At least I'm hoping that. 

I used to think maybe they were open at the top and caught rain water and stored it. But then there would be birds and stuff getting in there. And we're drinking that? Luckily how I thought they worked isn't at all how they work. You can read last weeks blog for some basic info on how they do in fact work. 

Here is a great example of a math application. It is a combination cone and cylinder. I tried to blow it up so you can see the numbers. Of course you can simply make up problems with numbers, but its nice to have numbers of an actual thing - even if the thing is just a picture from the internet.

Our town has a cylindrical water tower whose base sits on the ground. A good project would be to estimate the number of gallons it would hold. You could estimate the the diameter by first pacing off the circumference and then doing a little math. Then you could just estimate the height by eyeballing it, or better yet, doing some trigonometry. Granted, it would be a pretty rough estimate, but a nice project. I'm sure the water department, or someone, has the actual numbers. You could then get those and compare your estimate to what they say. 

I saw a company on-line that said they had towers, "Available in diameters from 11 feet (3.3 m) to 204 feet (62.2 m) and capacity from 20,000 gallons (75 cu m) to over 6 million gallons (22,700 cu m)". They are the self-proclaimed "premium water and liquid storage technology leader", so they must know their stuff.

This is a picture from their website. Oddly they don't mention the height, So a question might be:  For a given diameter, say 50 feet, what height would be necessary to have a 100,000 gallon tank? I looked up the fact that there are 7.48 cubic feet in a gallon. There you go. A great application.

A look on line shows that there are certain fairly standard shapes, but quite a few atypical designs, lending themselves to using several different volume formulas.

Water Towers

Do you know how water towers work? I have never known. I asked some people recently and they didn't know either. They're filled with rain water? It looks like a lot of water, but not nearly enough to last a city for very long. What happens if/when they run out? So many questions.

Here is a crash course that isn't going to cover everything, but it may answer a few questions you might have.

I thought the process kind of starts at the water tower, but it doesn't. The water we drink starts off at a water treatment plant. In a college biology class, we had a field trip to one. It was my only college field trip ever. Since I have all these questions now, I must not have paid very good attention.

Depending on the need, water is then pumped to the city or up into the tower. The water in the tower is just there in case it is needed. It holds a lot, about 50 swimming pools full, but still wouldn't last a city very long. It is probably enough for about a day. It can be used if power goes out, which would shut down the pumps. It can also be used for emergencies, such as fighting a fire. It is also used as a supplement if what is being pumped from the water treatment plant isn't enough.

As mentioned, it can be used in the case of a power outage. That is because water comes out from the tower by the force of gravity. That is why they are always located on the top of a hill. The water inside can only flow to points at a lower elevation than the tower itself.

There are plenty of good math applications involved here. Obviously, finding the capacity of the tower can be important. As can be seen, they might be conical, spherical, or cylindrical, or even combinations of shapes.

Finding the amount of pressure of the water flowing from the tower would be important. Every foot in height supplies 0.43 pounds per square inch. Thus, the higher it is, the more pressure. We'll look at these issues a little more next time.

Until now, I obviously never took the time to look into this topic. I got a lot of my information from several sources, but http://people.howstuffworks.com/water.htm was probably the best. That is where the diagram below is from.

A = From Treatment Plant
B = Pump
C = Water Tower
D = To Customers

The Bermuda Triangle

The Bermuda Triangle. It is the site of many shipwrecks and plane disappearances. Many dispute whether this is really all that extraordinary. After all, as can be seen, it encompasses a large area. Plus it is an area that has a lot of hurricanes and other disturbances in the weather. So all the mishaps could be something weird going on or maybe not that strange at all.

As mentioned, it is a large area. How large in fact. We have the standard A = 0.5bh formula, but it isn't easily used for this. So, this is a great application of Heron's formula.

To review, for any triangle with sides a, b, and c; what is called its semiperimeter is s = (a+b+c)/2.

The area then is A = square root of (s(s-a)(s-b)(s-c))

So, lets try it out. The value of s turns out to be 1,511. Plugging the numbers for a, b, c, and s into Heron's Formula gives a value of 437,600 square miles.

Different people give differing locations for the sides and vertices of the triangle, so there will be some differences. Also, we are not taking into account the curvature of the earth. Even so, it's a nice application.

Vietnam Memorial

The are many structures in the world that can be used to generate math applications. One of these is the Vietnam Memorial in Washington D.C. 

In 1980, individuals were invited to submit possible designs for the memorial. Over two thousands plans were submitted. The one selected was designed by Maya Ying Lin - at the time, an undergraduate architecture student at Yale University. She is currently an accomplished artist and designer. The wall was built in 1982. 

Its design is such that one wing points directly to the Washington Monument and the other to the Lincoln Memorial. At its greatest height, it stands 10 feet 3 inches. The top is at ground level, so the memorial itself actually sits below ground. Each wall is 246 feet 9 inches long. The walls meet at an angle of 125°12′. What look to be giant triangles are actually trapezoids with a long base of 10 feet three inches tapering to a small bases of 8 inches at each end.   

There are a number of math problems possible ranging from simple geometry to trigonometry. 

  

- How far is it from one end to the other? (Use the angle, sides of 246 feet, 9 inches and the law of cosines.)

- What is the area of the memorial? (Use the formula for area of a trapezoid.)

- What would be the length of one section of the memorial if extended to make a triangle? (This baffled me for a while. You can set up a proportion from the sides of the triangles that are formed to find a solution.)

Measuring Marajuana

I heard on the radio that someone was arrested that had 127 pounds of marijuana in his car. I wondered if that was even possible. Of the little I know about it, it is really light, so 127 pounds worth would probably take up a lot of space.

So what might that look like? Would it fill a shoe box? A trunk? The entire car?

Note: If you do this exploration with your math class, you probably need to be careful in how you frame this. If you present this as doing this as research as a member of law enforcement, you should be fine.

Anyway, I started my quest by going on-line. I found some websites that seemed to be in the know. The first person I read about had pretty much my same question regarding the relationship between volume and weight of marijuana.

His was a short post - about five sentences. I counted 48 typos. Not good, - e.g. "i duno wher im goin. somone help me out tho is there a way i cud mesure cannabis with a mesuring cup". I say about 5 sentences because punctuation did not seem to be his strong suit. On the plus side, he does know how to correctly spell cannabis.

A lot of people said this all depends on a number of different factors. I'm sure it does, but all I'm looking for is an estimation. One said an ounce is about a ziplock bag full. Another said he got an ounce that was about 3"x 3"x 2". They seemed like they were in the same ballpark, so lets go with that.

• So, a one ounce weight is 3"x 3"x 2" = 18 cubic inches.
• There are 16 ounces in a pound, so 1 pound is (16x18 =) 288 cubic inches
• 127 pounds must be (127x288 =) 36,576 cubic inches
• This would be easier for me to picture in cubic feet. There are 12x12x12 = 1728 cubic inches in a cubic foot, so  36,576 / 1728 = 21.17 cubic feet.
• The cube root of our answer is 2.8

So a box that is 2.8 feet on each side would hold this guy's marijuana. So you might be able to get that in your trunk. It certainly fits in the back seat of your car, with plenty of room left over for your other crime paraphernalia. But I don't know if you want that in your back seat. Maybe it didn't fit in his trunk, so he put it in plain sight in his back seat and that is what caused him to make the news.

Vitruvian Man II

 Last week I brought up some facts on ratios in the human body and how that relates to areas such as professional basketball. They weren't my facts, but from the book The Sports Gene.

This has to do with Leonardo da Vinci's sketch of The Vitruvian Man. That made me curious about that topic and how it got that for a name. I did a little research and here, what I think, are a few  are interesting tidbits on this topic.

• The drawing gets its name from Vitruvius, who was an accomplished architect in the Roman Empire, born approximately seventy years before Christ.
• Vitruvius made no drawings of his man, but described him.
• Several Italians including da Vinci made drawings based on Vistruvius' description, but not until 1,500 years later.
• Although others made drawings, da Vinci was the first to overlay the circle and square to show relationships.
• Leonardo was born in 1452 in Vinci, Italy - Thus "da Vinci".
• Leonardo's drawing was in one of his notebooks, which is why there is writing above and below the drawing.
• Even though it was on paper, the drawing still exists. It is located in Venice and is occasionally displayed.
• As mentioned in The Sports Gene, a person's wingspan is not always the same as the height (although mine is to the inch). If interested, there are several good class projects involving measurement and The Vitruvian Man on the internet. 
• Leonardo of course was famous for other works of art as well. I'm guessing that this along with The Last Supper and The Mona Lisa make him the artist with the most parodied works ever. (See below for a small sampling).

Vitruvian Man

I read an interesting book recently called, The Sport Gene, written by  David Epstein. Why do some excel in athletics and some don't - even with seemingly equal amounts of training? I always have liked to think that effort made all the difference. Anyone could be a great musician, a great high jumper, a great quarterback if one is just willing to put in the work.

Work is part of the equation for certain, but it turns out it's not the whole thing by any means. Some are just born with certain advantages.

Ted Williams was a great baseball player and some think maybe the best hitter of all-time. I'm sure he worked at hitting, but he didn't work at having 20/10 vision, which seemed to be a major help to him.

Muhammad Ali "reacted to light in 150 milliseconds, near the theoretical limit of human visual reaction time". Again, this is probable something innate and not developed.

The head of a athletic performance center said, "We've tested over ten thousand boys, and I've never seen a boy who was slow become fast."

What does it take to become an NBA player. One thing that helps your chances is height. The average NBA player is 15% taller than the average male. But it's more complicated than that.

The Vitruvian man is a famous drawing by Leonardo DaVinci. One can see that by Leonardo's placement of the man in the square is his assumption that a man's arm span is the same as his height. Is this always true? A class might try taking measurements and see if there is indeed a ratio of 1:1. Likely it will come quite close to that. I did my own measurement - exactly 1:1.

The book makes the point that in the 2010-11 season, there were only two players under that ratio. NBA players have an average ratio of 1.063:1. One of the current stars of the league is Anthony Davis. He is tall - 6 feet 9.75 inches, but his arm span is 7 feet 5.5 inches. That is a ratio of 1.095. He is tall, but effectively taller than his height would suggest.

Area of Colorado II

A blog from a couple weeks ago has bothered me. If we can learn from our mistakes, then I have an opportunity here.

I thought I would try finding the area of Colorado. It is a rectangle. That was mistake #1. It is bordered by lines of longitude of 102W and 109W. That is a difference of 7 degrees. but distances between lines of longitude don't stay the same. The distance between them narrows as you move from the equator going toward the poles where they all meet. So Colorado is really more of a trapezoid. And there we have mistake #2. It really isn't a trapezoid since it is on the surface of a sphere. So there are non-Euclidean aspects to deal with. I won't be dealing with that as I'm having enough problems with this. I'll stick with Euclid.

In mistakes #3 and counting, I found that it either can't be done, or I can't figure out how to get those horizontal boundary distances. I resorted to a website where you can plug in latitudes and longitudes and it will compute the distance using something called the haversine formula.

Long story short, doing so told me the south boundary is 386.2 miles and the north boundary is 363.9 miles. A couple weeks ago I had correctly computed the height as lines of latitude are parallel and thus stay the same distance apart. That value was 276.4 miles. Using the formula for area of a trapezoid, I got 103,056 square miles. The internet says it is 103,718 square miles. I'm calling that a win.

I'm not sure how they actually figure the areas of states. Maybe its just an estimate. Online I found at least three different values for the area of the state on sites that seemed to be fairly credible. So, my guess is as good as theirs.

Area of Colorado

I thought it might be interesting to try to compute the area of Colorado by using arc lengths - just to see if it comes out right. It doesn't. At least not how I did it. Maybe its of use to someone that can learn from my mistake(s).

One troubling thing is that I looked on the internet and got three different values for the area of the state. I would think in the age of GPS we would have that figured out to a pretty precise amount. The three amounts were separated by 94 square miles. At least that gives me some wiggle room.

I figured I could find both the length and width of the state would be with:

                                          (arc length)/360(2pi(radius of the earth))

The radius of the earth is 3,959 miles. The arc length I figured would be the differences in the latitudes or the longitudes. The height came out great. I got 276.4, and the web says 276. The width wasn't even close: my 483.7 to their 387. Mine was too big. Arc length would include the curvature of the earth. I thought maybe I could use Law of Cosines to get a closer figure.

                                C^2 = (3959)^2+(3959)^2-2(3959)(3959)Cos(7)

It was interesting that I got 483.4 as compared to an arclength of 483.7. What was not interesting was the web says the length of Colorado is 387.

What I learned:

 1.  I noticed their published lengths and widths didn't multiply to get their published area. It was 104,091 to 106,812 square miles. So, they must not use length x width to get the area. That makes sense, because that would only work on rectangular states of which there are not many.

2. Why my method didn't work. I knew this but didn't think about it - lines of longitude are not the same distance apart. They are a certain distance apart at the equator and shrink to nothing at the poles. I guess that is why my north-south distance came out accurately but my east-west was way off.

So, I don't know how they find the area of a state or any region for that matter.
There's a good project for you or your students.

Torus Formulas

I've had a week to reflect on the torus. Most of my reflections have been in the past few minutes leading up to my current act of typing, but I like to think my subconscious has been mulling it over.

Remember that last week we learned that the plural of torus is tori. True, even though my spell checker has put a squiggly red line under it.

I did think of another application. Remember a torus is a donut shape. Inner tubes of tires are also tori. So the square inch amount of rubber in the tire can be found with the surface area formula and the air in the tire would be found with the volume formula.

Also, there are apparently applications I would never have come up with.

In our model of cosmometry, the torus is the fundamental form of balanced energy flow found in sustainable systems at all scales. It is the primary component that enables a seamless fractal embedding of energy flow from micro-atomic to macro-galactic wherein each individual entity has its unique identity while also being connected with all else. 

This is from the website http://cosmometry.net/the-torus---dynamic-flow-process. I do not know what this all means. I'm not even sure what all the individual words mean, but it certainly sounds very important.

Last time I listed the formulas for the torus. I found there are other ways to find volume and surface area. Instead of using the variables in the way we used last week, these formulas use r, the distance from the center of the torus to the inner edge and R the distance from the center to the outer edge. 

V =  1/4(pi)^2(r+R)(R-r)^2

S.A. = (pi)^2(R^2-r^2)

(Again, I apologize for my inability to write exponents any other way.)

Beside finding volumes and surface areas of donuts, inner tubes, and various balanced energy flows (?) there is another nice application here. To find the ratio of surface area to volume of any three-dimensional shape is an important concept. To do so with the above formulas is especially cool as it simplifies down a lot.

Donut / Torus

I thought donuts would be an interesting topic. They are actually the mathematical shape called a torus. I first heard of this sitting in an undergraduate math class. Our professor told us we might try to find out about the torus before the next class. I actually looked it up. I was the only one in the class to do it and was able to talk about it the next time we met. I'm sure I got labeled as a nerd at that point. I wouldn't mind that, but when a room full of mathematicians think you're a nerd, that is probably an especially bad sign.

It turns out there is a lot I didn't know about this topic. Like the plural of torus. (It's tori, with a long i sound.) Is it donut or doughnut? (The consensus by those that decide these things seems to be "doughnut" although they seem to put up with "donut". "Donut really didn't come into regular use until Dunkin' Donuts started up in the 1950's.) I thought maybe this had a tie-in to the car, but no. It is spelled "Taurus" and I'm guessing has to do with the zodiac sign.

Imagine two circles linked as a chain. If one makes a full lap following the path of that first circle, we have a torus. Let's say the moving circle is radius r and the stationary circle has radius R.

The surface area is S = 4(pi^2)Rr. (Sorry, I do not know how to make my blog write the pi symbol or how to do exponents.) The derivation of this formula is more easily seen if written S = (2(pi)r)(2(pi)R). It is the circumference of the moving circle taking the a path along the circumference of the big circle.

The volume is V = 2(pi^2)(r^2)R. While this is the simplified version, again it is easier to see where it comes from by writing it differently: V = ((pi)(r^2))(2(pi)r). It is the area of the moving circle again taking the a path along the circumference of the big circle.

What can we use these formulas for? Not important, but there are a lot of them - donuts. Important, but none actually exist - the space station shown in the movie 2001. In the picture notice that it seems more of a rectangle than a circle on the outer edge. I think that is still a torus. The torus definition from different sources I found say "a closed curve", "a closed curve, especially a circle", or simply "a circle".

Enough for now. We'll look into this topic more next time.

Minature Football Field

A little over a year ago I was at the Football Hall of Fame in Canton, Ohio. In a somewhat related note, Canton is also the home and burial place of President William McKinley. The inside of the hall was awesome. But outside had a miniature artificial turf football field!! I think it was 40 yards long. For the most part, my use of my lawn consists of watering it and mowing it. I thought how awesome it would be to turn it into something like that. I figured that could be kind of expensive. I looked online and there was and ad for 10x10 feet of artificial turf for $95. Maybe its doable after all. At the very least, it makes for a cool math problem.

Suppose I want a field of 30 yards. If you are just messing around with folks you don't want to be running 100 yards to score. I don't know if my yard is 30 yards long, but let's say it is. Now, how wide should it be? The real deal is 53 1/3 yards. So mine should be the solution to

100 : 53 1/3 = 30 : x

It actually works out cleaner solving it with fractions. Anyway, x = 16 yards.

For the area, I have 30x16 = 480 square yards. However, I need this in square feet. There are 9 square feet in a yard, so 480x9 = 4,320 square feet. This could get a little expensive.

The cost, y would be found with 100 : 95 = 4,320 : y

So, y = $4,104. Also, I would have to somehow get yard markers, etc painted it. Still, it might be worth it.

Below is a picture of the field in Canton. (Hard to get a good picture from ground level.)